The Fourier-space <cosmological Poisson equation> gives
$$
\phi(\mathbf k,\tau)
=-\frac{4\pi Ga^2\bar\rho_m}{k^2}\delta_m
=-\frac{3\Omega_{m,0}H_0^2}{2ak^2}\delta_m.
$$
Insert this into the line-of-sight expression for the <CMB lensing potential>, Fourier transform $\delta_m$, and use the <Rayleigh plane-wave expansion>. Projection onto $Y_{lm}$ and spherical-harmonic orthogonality give
$$
\boxed{
a_{lm}^\psi=12\pi\Omega_{m,0}H_0^2i^{-l}
\int_0^{\chi_e}d\chi
\int\frac{d^3k}{(2\pi)^3}
\frac1{k^2}\frac{\chi_e-\chi}{\chi\chi_e}
\frac{\delta_m(\mathbf k,\tau)}{a(\tau)}
j_l(k\chi)Y_{lm}^*(\widehat{\mathbf k})}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 310 / 3 / c / Solution

The linearized <cosmological continuity equation>, the divergence of the <cosmological Euler equation>, and the <cosmological Poisson equation> give
$$
\boxed{\delta'+\theta=0},
\qquad
\boxed{\theta'+\mathcal H\theta
+\frac32\mathcal H^2\Omega_m\delta=0}.
$$
Eliminating $\theta=-\delta'$ yields the equation for a <linear cosmological density perturbation>:
$$
\boxed{\delta''+\mathcal H\delta'
-\frac32\mathcal H^2\Omega_m\delta=0}.
$$
In an <Einstein-de Sitter universe>, $\Omega_m=1$ and $\mathcal H=2/\tau$, so
$$
\delta''+\frac2\tau\delta'-\frac6{\tau^2}\delta=0.
$$
Substitution of the <power-law ansatz> $\delta\propto\tau^p$ gives $(p-2)(p+3)=0$. Hence
$$
\delta=C_+\tau^2+C_-\tau^{-3}
=D_+a+D_-a^{-3/2}.
$$
The leading one of the <matter-era growing and decaying density modes> is $\delta\propto a\propto\tau^2$, as direct substitution confirms, and its velocity divergence is
$$
\boxed{\theta=-\delta'=-\mathcal H\delta\propto-\tau}.
$$

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 312 / 4 / ii / Solution

On subhorizon scales, the Newtonian potential sourced by the matter density contrast obeys
$$
\nabla^2\phi=\frac32\mathcal H^2\Omega_m\delta.
$$


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 Incoming links: Cosmological Poisson equation